Discrete differential geometry is an energetic mathematical terrain the place differential geometry and discrete geometry meet and engage. It presents discrete equivalents of the geometric notions and techniques of differential geometry, akin to notions of curvature and integrability for polyhedral surfaces.

Additional info for A blend of methods of recursion theory and topology: A П 0^1 tree of shadow points

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But in order to do so, we need to integrate over a two dimensional region. So let U be some open subset of M and let ψ : U → π −1 U ⊂ O be a map satisfying π ◦ ψ = id. So ψ assigns a frame to each point of U in a differentiable manner. Let C be a curve on M and suppose that C lies in U . Then the surface determines a ribbon along this curve, namely the choice of frames from which e1 is tangent to the curve (and pointing in the positive direction). So we have a map R : C → O coming from the geometry of the surface, and (with now necessarily different notation from the preceding section) R∗ Θ12 = kds is the geodesic curvature of the ribbon as studied above.

Dt t If ω is a linear differential form, then we may compute i(Y )ω which is a function whose value at any point is obtained by evaluating the linear function ω(x) on the tangent vector Y (x). Thus i(φ∗t Y )φ∗t ω(x) = dφ∗t ω(φt x), dφ−t Y (φt x) = {i(Y )ω}(φt x). In other words, φ∗t {i(Y )ω} = i(φ∗t Y )φ∗t ω. We have verified this when ω is a differential form of degree one. e. a function, since then both sides are zero. But then, by the derivation property, we conclude that it is true for forms of all degrees.

If Z = YC is the restriction of a vector field Y to C we can define its “derivative” Z , also a vector field along C by YC (t) := ∇C (t) Y. 6) If g is a smooth function defined in a neighborhood of the image of C, and h is the pull back of g to I via C, so h(t) = g(C(t)) then the chain rule says that h (t) = d g(C(t)) = C (t)g, dt the derivative of g with respect to the tangent vector C (t). 2) implies that (hZ) = h Z + hZ . 6) hold. Indeed, to prove uniqueness, it is enough to prove uniqueness in a coordinate neighborhood, where Z j (t)(∂i )C .